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Mathematics > Analysis of PDEs

Title:A proof of Friedman's ergosphere instability for scalar waves

Abstract: Let $(\mathcal{M}^{3+1},g)$ be a real analytic, stationary and asymptotically
flat spacetime with a non-empty ergoregion $\mathscr{E}$ and no future event
horizon $\mathcal{H}^{+}$. On such spacetimes, Friedman provided a heuristic
argument that the energy of certain solutions $\phi$ of $\square_{g}\phi=0$
grows to $+\infty$ as time increases. In this paper, we provide a rigorous
proof of Friedman's instability. Our setting is, in fact, more general. We
consider smooth spacetimes $(\mathcal{M}^{d+1},g)$, for any $d\ge2$, not
necessarily globally real analytic. We impose only a unique continuation
condition for the wave equation across the boundary $\partial\mathscr{E}$ of
$\mathscr{E}$ on a small neighborhood of a point $p\in\partial\mathscr{E}$.
This condition always holds if $(\mathcal{M},g)$ is analytic in that
neighborhood of $p$, but it can also be inferred in the case when
$(\mathcal{M},g)$ possesses a second Killing field $\Phi$ such that the span of
$\Phi$ and the stationary Killing field $T$ is timelike on
$\partial\mathscr{E}$. We also allow the spacetimes $(\mathcal{M},g)$ under
consideration to possess a (possibly empty) future event horizon
$\mathcal{H}^{+}$, such that, however,
$\mathcal{H}^{+}\cap\mathscr{E}=\emptyset$ (excluding, thus, the Kerr exterior
family). As an application of our theorem, we infer an instability result for
the acoustical wave equation on the hydrodynamic vortex, a phenomenon first
investigated numerically by Oliveira, Cardoso and Crispino. Furthermore, as a
side benefit of our proof, we provide a derivation, based entirely on the
vector field method, of a Carleman-type estimate on the exterior of the
ergoregion for a general class of stationary and asymptotically flat
spacetimes.